Acta mathematica scientia,Series A ›› 2009, Vol. 29 ›› Issue (6): 1689-1698.

• Articles • Previous Articles     Next Articles

Almost Sure Central Limit Theorem for the Product of Partial Sums

  

  1. 1.Department of Mathematics, Soochow University, Jiangsu Suzhou 215006;
    2.Department of Mathematics, Zunyi Normal College, Guizhou Zunyi 563002
  • Received:2007-12-03 Revised:2009-08-26 Online:2009-12-25 Published:2009-12-25
  • Supported by:

    重庆市首批高等学校优秀人才支持计划项目(120060-20600204)资助

Abstract:

Let {Xn, n ≥ 1} be a sequence of i.i.d positive random variables with E(X1)=u >0, Var(X1)=σ2 and E|X1|3<∞, Sn=∑Nk=1Xk. Denote γ=σ/u the coefficient of variation.
 The authors give an unbounded measurable function g to satisfy the almost sure central limit theorem, i.e.,
 limN→∞1/logN∑Nn=11/n g((∏nk=1Sk/n!un )1/γ√n )=∫0g(x)dF(x), a.s.,
where F(•) is the distribution function of the random variable e√2 ξ and ξ is a standard normal random variable.

Key words: Almost sure central limit theorem, Partial sums, Unbounded measurable function

CLC Number: 

  • 60F15
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